Question:

The following table is a transportation matrix showing the cost ($) per unit of shipping from Plant i to Distributor j. The objective is to minimize shipping costs while meeting the demand placed by each Distributor. There is no shipment between Plant 2 and Distributor 3.

Distributor 1 Distributor 2 Distributor 3 Supply

Plant 1. 9 22. 24. 430

Plant 2. 14. 10 ..... 360

Plant 3. 18. 12. 13 220

Demand 400 390 190. ---

Let xij = number units shipped from Plant i to Distributor j

Solve the transportation problem and use the result to fill the boxes below.

All answers are equally weighted.

a) At optimal solution:

x11=

x12=

x13=

x21=

x22=

x23=

x31=

x32=

x33=

Minimum Cost = $

b) At optimal solution, how much unused capacity is available at each Plant?

Slack at Plant 1 = units

Slack at Plant 2 = units

Slack at Plant 3 = units

9. Suppose that a Markov chain has transition probability matrix 0 50 .50 0 P = .50 .25 .25 2 0 .50 .50 Give the values of limn-+x P(X, = i ) for i = 0, 1, 2. 10. Suppose that a Markov chain has transition probability matrix 0 1 2 .50 40 .10 P = .10 .50 .40 2 .40 .10 .50 Find the stationary probabilities To, 71, 12.\f2. Spectral Versus Singular Value Decomposition: Consider a linear operator A with matrix representation 1 3 A _ [7 _1] (a) Compute the spectral decomposition and singular 1mlue decomposition of A. (1:) Sketch ANSI}, the image of the unit circle under .A. (c) Compare your answers to parts (a) and {b} and draw a conclusion